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Subfield and field extension

Definition

Given a field $F$, the intersection of all subfields $S \subseteq F$ is a field. We call this the prime subfield of $F$.

Criterion

Properties

The prime subfield is isomorphic to either $\mathbb{Q}$ or $\mathbb{Z}/p\mathbb{Z}$. (Proof: extend the assignment $0 \to 0, \; 1 \to 1$ ring homomorphism $\mathbb{Z} \to F$).

Examples

Suppose $F$ is not algebraically closed and let $f(x) \in F[x]$ be an irreducible polynomial. Then (TODO)

all these pages adapted with probably insufficient credit from my university's lecture notes